Consider the following two infinite series $$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\, \sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}.$$ Denote the triangular numbers and pentagonal numbers by $t_n:=\binom{n+1}2$ and $\omega_n:=\frac{n(3n+1)}2$, respectively.

QUESTION. Are these two congruences true modulo $4$? $$\sum_{n\geq0}a(4n+3)q^n\equiv \sum_{n\in\mathbb{Z}}q^{5t_n} \qquad \text{and} \qquad \sum_{n\geq0}b(4n+3)q^n\equiv 2\sum_{n\in\mathbb{Z}}q^{14\omega_n}.$$