Rademacher's infinite partial fraction conjecture for partition generating functions

Let $p_N(n)$ denote the number of partitions of $n$ into at most $N$ parts, and write

$$\sum_{n\geq 0}p_N(n)x^n=\prod_{j=1}^N\frac{1}{1-x^j}$$

with partial-fraction coefficients $C_{h,k,l}(N)$ defined by

$$\prod_{j=1}^N\frac{1}{1-x^j}=\sum_{k=1}^N\sum_{\substack{0\leq h<k\gcd(h,k)=1}}\sum_{l=1}^{\lfloor N/k\rfloor}\frac{C_{h,k,l}(N)}{(x-e^{2\pi i h/k})^l}.$$

Here $s(h,k)$ is the Dedekind sum, $\Delta_\alpha$ is the forward difference operator, and $L_{3/2}$ is given by $L_{3/2}(-y^2)=-\frac{1}{2\sqrt{\pi}y^2}\left(2\cos(2y)-\frac{\sin(2y)}{y}\right)$. Rademacher's conjecture. For all integers $h,k,l$ such that $0\leq h<k$, $\gcd(h,k)=1$, and $l\geq1$, the limit $\lim_{N\to\infty}C_{h,k,l}(N)$ exists and equals

$$R_{h,k,l}:=-2\pi\left(\frac{\pi}{12}\right)^{3/2}\frac{e^{\pi i(s(h,k)+2hl/k)}}{k^{5/2}}\Delta^{l-1}_{\alpha}L_{3/2}\left(-\frac{\pi^2}{6k^2}(\alpha+1)\right),$$

evaluated at $\alpha=\frac{1}{24}$, where

$$s(h,k)=\sum_{\mu=1}^{k-1}\left(\frac{\mu}{k}-\left\lfloor\frac{\mu}{k}\right\rfloor-\frac12\right)\left(\frac{h\mu}{k}-\left\lfloor\frac{h\mu}{k}\right\rfloor-\frac12\right)$$

and

$$\Delta_\alpha^j f(\alpha)=\sum_{h=0}^j(-1)^h\binom{j}{h}f(\alpha+j-h).$$

The conjecture was open for nearly four decades, but the paper presents overwhelming evidence that these limits do not exist: the sequences oscillate and attain arbitrarily large positive and negative values. Thus the conjecture is almost certainly false.