Hikami's radial-limit conjecture for false theta functions

For $n\geq 0$, let $(x;q)_n=(1-x)(1-xq)\cdots(1-xq^{n-1})$ be the $q$-Pochhammer symbol, and let $\begin{bmatrix}n\k\end{bmatrix}_q$ denote the $q$-binomial coefficient. For $m\geq 2$ and $0\leq a\leq m-2$, define

$$\widetilde{\Phi}_m^{(a)}(q)=mq^{\frac{(m-a-1)^2}{4m}}\sum_{n_1,\dots,n_{m-1}\geq 0}(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+n_1^2+\cdots+n_{m-2}^2+n_{a+1}+\cdots+n_{m-2}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\ _i\end{bmatrix}_q.$$

It has the false-theta expansion

$$\widetilde{\Phi}_m^{(a)}(q)=m\sum_{n\geq0}\chi_{2m}^{(a)}(n)q^{n^2/4m},$$

where $\chi_{2m}^{(a)}(n)=1$ if $n\equiv m-a-1\pmod{2m}$, $-1$ if $n\equiv m+a+1\pmod{2m}$, and $0$ otherwise. Define

$$Y_{m,N}^{(a)}(q)=\sum_{n_1,\dots,n_{m-1}=0}^{N-1}(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+n_1^2+\cdots+n_{m-2}^2+n_{a+1}+\cdots+n_{m-2}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\ _i\end{bmatrix}_q.$$

Let $\zeta_N=e^{2\pi i/N}$. Hikami's conjecture. For any $m\geq2$ and $0\leq a\leq m-2$, the radial limit satisfies

$$\lim_{q\to\zeta_N}\widetilde{\Phi}_m^{(a)}(q)=\zeta_N^{\frac{(m-a-1)^2}{4m}}Y_{m,N}^{(a)}(\zeta_N).$$

False theta functions have well-defined radial limits at roots of unity, and these limits give quantum modular forms. The paper proves this conjecture, so its status is solved.