Subpandigital and subpenholodigital square existence conjecture

Let $b>2$. A strict subpandigital square or strict subpenholodigital square in base $b$ mean the square of a number whose base-$b$ representation contains, respectively, each digit $0,1,\ldots,b-2$ exactly once and no digit $b-1$, or each digit $1,\ldots,b-2$ exactly once and neither $0$ nor $b-1$.

Subpandigital and subpenholodigital square existence conjecture. Suppose $b>7$. A strict subpandigital square and a strict subpenholodigital square in base $b$ exist if and only if $b$ is even or $b-1$ has an odd $2$-adic valuation.

The preceding corollary rules out both types when $b$ is odd and $b-1$ has even $2$-adic valuation. The supplied text gives no resolution of the asserted existence in the complementary bases.