Three mod-$9$ identities from the $\mathrm{D}_4^{(3)}$ principal-subspace sum

Let $q$ be a formal variable, let $m_1,m_2\geq 0$, and define the theta-product notation by $\theta(a_1,\ldots,a_r;Q)=\prod_{i=1}^r(a_i;Q)_\infty(Q/a_i;Q)_\infty$. The mod-$9$ identities. The following three identities are conjectured:

$$\sum_{m_1,m_2\geq 0}\frac{q^{m_1^2+3m_1m_2+3m_2^2}}{(q;q)_{m_1}(q^3;q^3)_{m_2}}=\frac{1}{\theta(q,q^3;q^9)},$$ $$\sum_{m_1,m_2\geq 0}\frac{q^{m_1^2+3m_1m_2+3m_2^2+m_1+3m_2}}{(q;q)_{m_1}(q^3;q^3)_{m_2}}=\frac{1}{\theta(q^2,q^3;q^9)},$$ $$\sum_{m_1,m_2\geq 0}\frac{q^{m_1^2+3m_1m_2+3m_2^2+2m_1+3m_2}}{(q;q)_{m_1}(q^3;q^3)_{m_2}}=\frac{1}{\theta(q^3,q^4;q^9)}.$$

These identities arise from the principal-subspace character of the basic $\mathrm{D}_4^{(3)}$ module and were presented in the source as conjectural results. The supplied status gives no resolution evidence, so they remain open here.