Andrews's modified partition identity for $A_{4,3,3}(n)$

Let $A_{4,3,3}(n)$ be the partition-counting function defined by the conditions for even $\lambda=4$, and let $B_{4,3,3}(n)$ count partitions satisfying the stated difference and multiplicity conditions, where $f_j$ denotes the number of appearances of $j$. Define $B^0_{4,3,3}(n)$ by imposing in addition

$$f_{5j+2}+f_{5j+3}\leq 1\quad (j\geq 0),$$ $$f_{5j+4}+f_{5j+6}\leq 1\quad (j\geq 0),$$ $$f_{5j-1}+f_{5j}+f_{5j+5}+f_{5j+6}\leq 3\quad (j\geq 1).$$

Andrews's conjecture. For every positive integer $n$,

$$A_{4,3,3}(n)=B^0_{4,3,3}(n).$$

The conjecture gives a modification of the partition conditions in Andrews's theorem that admits the case $k<\lambda$; the source presents it as a conjecture attributed to Andrews. The paper's title indicates that this particular identity is subsequently proved by an analytic argument.