Closure conjecture for products of N=3 character spaces

Let $m,m'\in\mathbf N$ and $j,j'\in\mathbf Z$. For $m\in\mathbf N$ and $j\in\mathbf Z$, let $\overset{N=3}{CH}{}^{(+)[K(m),j]}$ denote the $\mathbf C((q^{\frac12}))$-linear span of the N=3 characters ${\rm ch}^{(+)}_{H(\Lambda^{[K(m),m_2]})}(\tau,z)$ with $m_2\in j+2\mathbf Z$ and $0\leq m_2\leq m$. Product-closure conjecture. The following inclusion will hold for all $m,m'\in\mathbf N$ and $j,j'\in\mathbf Z$:

$$\overset{N=3}{CH}{}^{(+)[K(m),j]}\cdot\overset{N=3}{CH}{}^{(+)[K(m'),j']}\subset\overset{N=3}{CH}{}^{(+)[K(m+m'),j+j']}.$$

Equivalently, for $m_2,m_2'\in\mathbf Z$ satisfying $0\leq m_2\leq m$ and $0\leq m_2'\leq m'$, there should exist functions $b^{(m,m_2),(m',m_2')}_{m_2”}\in\mathbf C((q^{\frac12}))$ such that

$${\rm ch}^{(+)}_{H(\Lambda^{[K(m),m_2]})}(\tau,z)\,{\rm ch}^{(+)}_{H(\Lambda^{[K(m'),m_2']})}(\tau,z)=\sum_{\substack{m_2”\in m_2+m_2'+2\mathbf Z\\0\leq m_2”\leq m+m'}}b^{(m,m_2),(m',m_2')}_{m_2”}(\tau)\,{\rm ch}^{(+)}_{H(\Lambda^{[K(m+m'),m_2”]})}(\tau,z).$$

This predicts that products of character spaces are closed under multiplication, with the parameters $m$ and $j$ adding. The preceding proposition establishes several special product inclusions, while the general product formula remains conjectural.