Minimum expected number of 5-die rolls to simulate a 6-die

Context: If we only have a fair 5-sided die, we can emulate a fair 6-side die roll by rolling D5 twice. Out of 25 outcomes, we can map 24. If we get the 25th outcome, eg. (5,5), we just repeat the procedure. (I think it’s called rejection sampling?)

The expected number $E$ of rolls here is:

$$E = 2\times\frac{24}{25} + \frac{2+E}{25}$$

Hence

$$E = 25/12 = 2.08\overline{3}$$

Question:can we do better? I feel like we can’t, but I don’t know how to prove it.