DataListPlot[
With[{mn = Min@#[[All, 2]], dat = #},
{#1/4 // N, (#2 - mn), #3, #4} & @@@ dat
],
PlotRange -> All, Frame -> True, Axes -> False, PlotRange -> All,
Joined -> True,
PlotMarkerFunction -> {(
With[{x = #1, y = #2, err = #3, conv = #4},
{If[conv,
## &[Darker@Blue,
Disk[{x, y}, Offset[{$PointSize, $PointSize}] ]], ## &[
Darker@Red,
Rectangle[Offset[{-$PointSize, -$PointSize}/2, {x, y}] ,
Offset[{$PointSize, $PointSize}/2, {x, y}] ]]],