Second time axis count in weeks

I have a time series that count as dates (set timefmt "%d.%m.%Y"). I would like to have a second, x2 axies displaying the same data in weeks (set format x2 "%Ww"), so I wrote this script (below) but the boxxy errorbars drawing for x2y1 seem to be shifted. Why is that?

$test0<<EOD 
m1  01.06.2025   30.06.2025 
m2  10.06.2025   17.07.2025 
m3  20.06.2025   08.08.2025 
EOD

set term qt font "Carlito,12" butt antialias size 1440,360

set timefmt "%d.%m.%Y" 
OneMonth = strptime("%m","2")
set xdata time 
set format x "%b %y" 
set x2data time 
set format x2 "%Ww"

unset border

set yrange [-1:3] 
set xrange ["20.05.2025":"20.08.2025"] 
set xtics OneMonth nomirror 
set xtics offset 0,0.5 
set x2tics nomirror scale 0.8,0.5 font ",10" offset 0,-0.5 
set mxtics 4 set mx2tics 2 set ytics nomirror offset 1.5,0

set style fill transparent solid 0.3 
set key noautotitle 
set ytics nomirror 
set grid y 
unset key

plot    $test0 using 2:($0) : 2:3 : ($0)-0.2:($0)+0.2 : yticlabel(1) with  boxxy lc "blue" axes x1y1,\
        $test0 using 2:($0) : 2:3 : ($0)-0.2:($0)+0.2 : yticlabel(1) with  boxxy lc "green" axes x2y1

The x and x2 axis ranges are independent unless you do something to force them to match. That could either be setting them both to the same range in parallel:

set xrange  ["20.05.2025":"20.08.2025"] 
set x2range ["20.05.2025":"20.08.2025"]

or using the set link command to link them automatically:

set link x2