Subpandigital and subpenholodigital prime digit-sum conjecture

For a base-$b$ integer $n$, let $s_b(n)$ denote the sum of its base-$b$ digits. A subpandigital prime (respectively, subpenholodigital prime) is a prime whose base-$b$ representation contains each digit $0,1,\ldots,b-2$ (respectively, each digit $1,\ldots,b-2$) at least once, while omitting $b-1$ (respectively, omitting both $0$ and $b-1$).

Subpandigital and subpenholodigital prime digit-sum conjecture. For $b>4$, the smallest subpandigital prime or smallest subpenholodigital prime $n$ satisfies $s_b(n)=(b-2)(b-1)/2+2$ for $b$ of the form $4k+3$, and satisfies $s_b(n)=(b-2)(b-1)/2+1$ otherwise.

The preceding theorems provide lower bounds and improve them for bases of the form $4k+3$. The conjecture is motivated by the displayed numerical table, and the supplied text gives no resolution.