Both would and will are possible in that sentence, though neither is necessary

Will is felicitous if you believe they're still out there searching

What matters is your belief, not reality

You obviously know about the bank robbers from an external source of information because you said those bank robbers

It's not a stretch to imagine that the search is ongoing if you haven't heard news of their capture yet

But this all depends on context (hence "felicitous" rather than "grammatical")

@DamkerngT. That would probably need since yesterday

Infinitive complement of modal auxiliary

Felicitous = appropriate in context

Some things are grammatical, but may not always be appropriate

If you divide errors into grammatical, semantic, and pragmatic contexts, you get "ungrammatical", "nonsensical", and "infelicitous" respectively

Although in real life the three categories are probably overlapping

For example, think about a and the. Many sentences are grammatical regardless of which one you use, but in context usually only one or the other is appropriate.

So when you change would to will, you get a grammatical sentence, but it may not be appropriate in the same contexts.

@DamkerngT. There's a couple points of theory there:

1. Is the auxiliary verb a dependent of the main verb? Or is it the other way around?

You can say like takes a to-infinitival complement, while can takes a bare infinitival complement.

You don't have to treat auxiliary verbs as special in this sense.

2. The distinction between forms and the uses of those forms: I would say it's the plain form, much like Collins would say base form

But I would says it's the plain form used as an infinitive (in an infinitival construction)

You don't have to say that. I see no reason not to, though

Alternatively, you can consider infinitive itself a specific form

(As in traditional grammar)

@Nico I took them both as expressing uncertainty, but would as expressing past time as well

The B variants are similar to the A variants, but less certain--they're expressing fairly confident speculation, but the speaker doesn't know that what they're saying is correct, they just believe it's pretty likely (presumption)

So the modal auxiliaries there have to do with the speaker's lack of information

In other contexts, other meanings would be possible

Like, in another context, "I'm sure they will have been" could be a statement about a state continued until a point in the future

And in another context, "I'm sure they would have been" could be counterfactual (like, "I'm sure they would have been looking for those bank robbers [if they had any manpower to spare]")

And so on.

@Fantasier Me too. I usually avoid answering questions about them :-)

@Fantasier Sure. I think modal verbs are hard for most learners, although of course it depends on how similar it is to your L1

Though that goes for everything in language learning--no language is harder or easier than another in a vacuum, it's all relative to what language(s) you already speak

@DamkerngT. Think of how hard Thai would be for a speaker of Lao versus a speaker of English

Or Japanese for a speaker of Korean versus a speaker of French

The only language learning you do in a vacuum is L1, and everyone manages to learn their L1.

There are extralinguistic concerns that are harder in one language or another, like writing systems, for example

Keep with it. Write out the whole fifty-sounds chart every day until you can do it from memory. It's okay if your handwriting isn't super neat

I used to tell people to start by adding one row of kana each day: day one, write あいうえお, day two write あいうえお and かきくけこ, and so on. Keep doing it until you've got the chart so far memorized, then stop, add 5 new flashcards (on 3x5" note cards), and quiz yourself on recognition.

Some people try to cram all the kana in a day or two, or use Anki, or whatever method they like.

Ultimately it doesn't matter since there's only a few kana, and if you use them every day learning them shouldn't be much of an issue regardless of method, since you'll see them millions upon millions of times

(Methods for stuff like this matter more for students who have difficulty learning to read and write. Most students can make do with any old approach :-)

Yes, for a number of reasons