Perhaps:
1500uF, 6.3V, 8x15mm: https://www.mouser.co.uk/ProductDetail/Panaso … ic/EEU-FS0J152L
1000uF, 10V, 8x16mm (a little taller): https://www.mouser.co.uk/ProductDetail/Rubyco … ZLH1000MEFC8X16

Both low ESR, 105C, longish life. Both (as far as I know) reputable makes, I've used them before repairing a couple of boards with no problems. Mind that the 1000uF ones are 2mm taller than yours, but I couldn't immediately find any the original size. In my limited experience the smaller value caps don't tend to cause problems. Except for some surface mount electrolytics that leaked and broke a floppy drive I had, but I think that was a slow leak from a seal failing on much older capacitors rather than the capacitor bursting as happened during the capacitor plague times. I've got an Abit board from about the same time as the BE6 and that had problems with all of the large value caps (anything >1000uF), but no problems with the smaller ones.

snufkin wrote on 2021-11-14, 11:11:

Perhaps: 1500uF, 6.3V, 8x15mm: https://www.mouser.co.uk/ProductDetail/Panaso … ic/EEU-FS0J152L 1000uF, 10V, 8x16mm (a little tal […]
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Perhaps:
1500uF, 6.3V, 8x15mm: https://www.mouser.co.uk/ProductDetail/Panaso … ic/EEU-FS0J152L
1000uF, 10V, 8x16mm (a little taller): https://www.mouser.co.uk/ProductDetail/Rubyco … ZLH1000MEFC8X16

Both low ESR, 105C, longish life. Both (as far as I know) reputable makes, I've used them before repairing a couple of boards with no problems. Mind that the 1000uF ones are 2mm taller than yours, but I couldn't immediately find any the original size. In my limited experience the smaller value caps don't tend to cause problems. Except for some surface mount electrolytics that leaked and broke a floppy drive I had, but I think that was a slow leak from a seal failing on much older capacitors rather than the capacitor bursting as happened during the capacitor plague times. I've got an Abit board from about the same time as the BE6 and that had problems with all of the large value caps (anything >1000uF), but no problems with the smaller ones.

@snufkin , thanks for the links - I'll see about sourcing them in the next few days. Height won't be a problem on any of the 1500 / 1000 ones, but will need to stick to 8mm dia as they're tightly spaced on the board. Only have a couple other Abit boards and one of them was recapped a while back (not by me) with Rubycon ZLs.

To be honest I mostly picked out those because they're what Mouser had in stock and will sell in small quantities. If it makes sourcing easier, anything that's listed as low ESR of the right size from a reputable manufacturer (Panasonic, Rubycon, Nichicon, Wurth, UCC, I'm sure there are a couple I'm forgetting) should be fine. I like to go with 105C rated not because the temps will get that high, but it means the capacitor is working further away from its rated maximum, which extends its lifespan. I was told it's not linear, so doubling the difference between working and maximum temperature (e.g. working at 65C, rated for either 85C or 105C) will more than double the lifespan.

snufkin wrote on 2021-11-15, 11:20:

I was told it's not linear, so doubling the difference between working and maximum temperature (e.g. working at 65C, rated for either 85C or 105C) will more than double the lifespan.

The usual equation is it doubles every 10°C.😃👍