The capacitors in decent speaker crossovers are never
electrolytics, so they don't generally deteriorate over time.

Never? I haven't checked "decent" speakers lately, but the AR11 is from an
era when non-polar electrolytics were commonly used. Even inexpensive Mylar
caps cost more than electrolytics.

The tendency toward two-way designs (the AR11 is three-way) has raised the
crossover point to a frequency that makes the use of a film cap
more-practical. But even "the" classic two-way speaker (The Advent Speaker)
used an electrolytic.

The DQ-10 was one of the first speakers to use film caps. The original
version used mostly electrolytics; the changeover occurred several years
into production. In exchange for loaning him a pair of Pearl mics, Jon sent
me a box of cheap Mexican Mylar caps, and I repopulated the crossovers. I
was in such a rush to get it all done, that I didn't leave one of my four
DQ-10s unmodded, and wasn't able to make a valid comparison.

I /was/ going to suggest a mechanical problem with the woofer or midrange,
but the OP /specifically stated/ that the hissing sound had no relationship
with what was playing, or how loud it was, or /any movement of the cone/.