[maximin] geometric climbing

A puzzle from The Riddler this week returning to the ranking of climbing competitors in Tokyo. And asking for the maximin score, that is, the worst possible absolute score guaranteeing victory. In the case of eight competitors, a random search for a maximin over 10⁶ draws leads to a value of 48=1x7x8, for a distribution of ranks as follows

[1,] 1 7 5
[2,] 2 3 7
[3,] 3 4 3
[4,] 4 5 2
[5,] 5 2 4
[6,] 6 1 6
[7,] 7 6 1

while over seven competitors (the case with men this year, since one of the brothers Mawem got hurt during the qualification), the value is 35=1x5x7, for a distribution of ranks as follows

[1,] 1 7 7
[2,] 2 6 4
[3,] 3 5 3
[4,] 4 4 6
[5,] 5 2 5
[6,] 6 8 1
[7,] 7 1 8
[8,] 8 3 2

exhibiting a tie in the later case (and no first position for the winners!).