Large-loop asymptotic conjecture for the primitive beta function

Let $b2^{\mathrm{prim}}_L$ denote the contribution to the beta function from primitive vertex-type graphs at loop order $L$, and let $\bar\beta_L$ be the asymptotic expression displayed below, with $N$ the number of vector components, $A$ the constant appearing in that expression, and $\gamma_{\mathrm E}$ Euler's constant.

\bar \beta_L=\Gammaleft(L+\frac{N+10}{2}\right)\frac{36\cdot3^{\frac{N+1}{2}}}{\pi\Gammaleft(2+\frac N2\right)A^{2N+4}\right)e^{-\frac32-\frac{N+8}{3}\left(\frac34+\gamma_{\mathrm E}\right)}.

Primitive beta-function asymptotic conjecture. The leading growth of the primitive beta function satisfies

$$\beta^{\mathrm{prim}}_L\sim\bar\beta_L\left(1+\mathcal O\left(\frac1L\right)\right).$$

This is the conjectured coincidence between the leading asymptotic growth of the primitive beta function and that of the full beta function in minimal subtraction. The supplied text reports numerical results through 17 loops but gives no proof or resolution.