A-F-M conjecture on sums of reproducing kernels

Let $K_1$ and $K_2$ be reproducing kernels of the form

$$K_1(z,w)=\sum_{k\in\mathbb Z_+}a_k(z\bar w)^k,\qquad K_2(z,w)=\sum_{k\in\mathbb Z_+}b_k(z\bar w)^k.$$

Assume

$$\lim_{k\to\infty}\frac{a_k}{a_{k+1}}=\lim_{k\to\infty}\frac{b_k}{b_{k+1}}=1,$$ $$\lim_{k\to\infty}a_k=\lim_{k\to\infty}b_k=\infty,$$

and that there are measures $\nu_1$ and $\nu_2$ on $[0,1]$ such that, for every $k\in\mathbb Z_+$,

$$\frac{1}{a_k}=\int_{[0,1]}t^k\,d\nu_1(t),\qquad \frac{1}{b_k}=\int_{[0,1]}t^k\,d\nu_2(t).$$

A-F-M conjecture. The multiplication operator $M_z$ on $\mathcal H(K_1+K_2)$ is subnormal.

The conjecture concerns preservation of subnormality for multiplication operators associated with sums of positive definite kernels. The analogous question for products of kernels has been answered negatively in general, whereas the stated sum problem remains unresolved on the supplied evidence.