The representation-count formula for binary quadratic form numbers

Let a ${\mathcal B}$-number be a positive integer represented by the binary quadratic form $a^2+ab+b^2$. Suppose that such a number is written in the general form

$$n=x^2\cdot 3^y\cdot a^\alpha\cdot b^\beta\cdot c^\gamma\cdots,$$

where $x$ has no prime factor of the form $6k+1$, and $a,b,c,\ldots$ are primes of the form $6k+1$. The exponents $\alpha,\beta,\gamma,\ldots$ are nonnegative integers. Representation-count formula. The number of distinct ${\mathcal B}$-representations of $n$, including the cases $a=b$ and either variable being zero, is

$$\begin{cases} \frac{1}{2}\Bigl(1+(\alpha+1)(\beta+1)(\gamma+1)\cdots\Bigr), & \text{if all of }\alpha,\beta,\gamma,\ldots\text{ are even};\\ \frac{1}{2}(\alpha+1)(\beta+1)(\gamma+1)\cdots, & \text{otherwise.} \end{cases}$$

This gives an explicit divisor-exponent formula for the number of representations by $a^2+ab+b^2$, refining the preceding factorization results for ${\mathcal B}$-numbers. The source does not provide evidence resolving the claim, so its status is left open.