Just a WAG, but perhaps in normal operation the dim digits were constantly
lit, while the bright ones weren't?

could be, but I doubt it. The brightness starts going down from digit 10 and
it is really progressive, each digit is dimmer than the one on its
left.

That would suggest to me that the filament is being driven by DC,
rather than by AC or a square wave.

The filament is the cathode in the tube. The voltage difference
between the cathode, and the fluorescent anode sections, greatly
affects the brightness. If you have a voltage drop across the width
of the filament (with one side grounded) you'll have different
cathode-to-anode voltages across the width of the display, and
different digit brightnesses.

One way to avoid this (the usual way) is to heat the filament with an
AC or square-wave-DC signal, typically symmetrical around ground.
This results in the same average voltage between the anodes, and
different points on the filament, and evens out the brightness. If
you drive the filament with AC from a separate transformer winding,
it's necessary to create some sort of ground return reference for it
(for the cathode-to-anode electron flow). This can be done by
grounding a center tap on the transformer secondary, or grounding a
center point on the filament itself.

The fact that swapping the filament pins didn't affect the problem,
makes me suspect that there's a fault inside the tube. Possibly
there's an internal short between one end of the filament, and ground.

So, take a look at the voltages and waveforms on the filament drive
pins. Ideally, they should be exactly mirror-opposites of one
another, with respect to ground. I suspect you'll find that they are
not.