Bernoulli numerator divisibility and representation by a quadratic form

Let $p$ be a prime, let $B_{p-1}$ denote the $(p-1)$-st Bernoulli number, and let $\operatorname{num}(B_{p-1})$ denote its numerator. Consider the quadratic form

$$F(x,y)=-x^2+2xy+2y^2.$$

Bernoulli–quadratic-form conjecture. For every prime $p$,

$$\frac{p-1}{2}\mid \operatorname{num}(B_{p-1})\quad\Longleftrightarrow\quad p\text{ is representable by }F(x,y).$$

The claim connects divisibility properties of Bernoulli numbers with representation of primes by a binary quadratic form of discriminant $12$. The source presents it as a conjectural pattern, and no resolution is supplied here.