Teaser 3036: That old chestnut

From The Sunday Times, 29th November 2020 [link]

Clearing out an old drawer I found a wrinkled conker. It was my magnificent old 6709-er, a title earned by being the only survivor of a competition that I had had with friends. The competition had started with five conkers, veterans of many campaigns; each had begun at a different value between 1300 and 1400.

We used the rule that if an m-er beat an n-er in an encounter (by destroying it, of course!) the m-er would become an m+n+1-er; in effect, at any time the value of a conker was the number of destroyed conkers in all confrontations in its “ancestry”.

I recall that at the beginning of, and throughout, the competition, the value of every surviving conker was a prime number.

What were the values of the five conkers at the start?