Several of the tants did fail on my early 90s 34401A, two of the Zeners were damaged as a result, thankfully I spotted the AC & frequency measurement wasn't working before the PCB got burnt, the self test will identify these failures too, if you mostly use it for DC measurements.
After this I did replace all the tantalum caps & the electrolytics on the mainboard, leaving 30 year old radial caps to potentially pee electrolyte over the board, is not a risk I'm prepared to take (having changed many in the non TE related day job & other TE I own that had caused corrosion). They hadn't leaked in mine, YMMV.
Parts I used to replace the tants were from the Kemet (now Yageo) T489 low leakage range.
6x 22uF 35V (was 20V across 15V rails), T489D226K035ATE300
1x 4.7uF 25V (was 10V across 5V), T489B475K025ATE1K0
2x 47uF 25V (was 10V across 5V), T489D476K025ATE250
3x 1uF 35V (same rating, 18V rails), T489B105K035ATE2K0
Zeners I used were Onsemi BZX84B3V3LT1G
I also bought 3x more of the 1uF 35V, for the display board but decided against replacing them, due to them being under the VFD, didn't want to risk damaging it, there are up to 3 used depending on the age/revision.
After this the board got cleaned with flux remover & IPA several times.
There is a later (1999) 34401A here, I haven't replaced anything in that one, but ordered enough tants to keep a spare set ready, in case that fails in the future.
David