Affine theta-function expansion conjecture for the $A_{N-1}$ blow-up formula

Let $N$ be an odd prime. Let $A_{N-1}$ be the Cartan matrix, let $\tau$ be in the upper half-plane, and let $M$ and $M^*$ denote the root lattice and dual lattice of $A_{N-1}$, respectively. For a weight vector $\beta\in \frac{M^*}{NM}$, let $\theta_{A_{N-1}}^\beta(\tau)$ be a level-$N$ $A_{N-1}$ theta function.

Affine theta-function expansion conjecture. The $A_{N-1}$ theta function in the blow-up formula satisfies

$$\frac{1}{\eta(\frac{\tau}{N})}=\frac{\theta_{A_{N-1}}(\tau)}{\eta(\tau)^N},$$

and admits an expansion

$$\theta_{A_{N-1}}(\tau)=\sum_{\beta\in \frac{M^*}{NM}}n_\beta\theta_{A_{N-1}}^\beta(\tau),\qquad n_\beta\in\mathbb Z.$$

This proposes expressing the blow-up-formula theta function in terms of familiar level-$N$ theta functions; the paper verifies the claim for small rank, while its general validity is not established in the supplied text.