Flawlessness conjecture for rank-generating functions

Let $F_{\lambda}(q)=\sum_{i=0}^{d}c_iq^i$ be the rank-generating function for partitions contained inside a partition $\lambda$. The polynomial is flawless if $c_i\le c_{d-i}$ for every $i\le d/2$. Flawlessness conjecture. For every partition $\lambda$, the rank-generating function $F_{\lambda}(q)$ is flawless. The source presents this as an open conjecture and gives no general proof or counterexample.