Perfection monotonicity conjecture for two-rowed Specht ideals

Let $n$ and $k$ be integers satisfying $1\leq k<k+1\leq n-k$, and let $\mathfrak{a}(n+1,j+1,j+1)$ denote the Specht ideal associated with the partition $(n+1,j+1,j+1)$. Perfection monotonicity conjecture. If

$$\mathfrak{a}(n+1,k+1,k+1)$$

is perfect for some integer $k$ satisfying $1\leq k<k+1\leq n-k$, then

$$\mathfrak{a}(n+1,i+1,i+1)$$

is perfect for all integers $i$ satisfying $1\leq i\leq k$. This conjecture concerns the propagation of perfection among the relevant two-rowed Specht ideals; the source presents it as equivalent to the claim that $\mathfrak{a}(n+1,k+1,k+1)$ is not perfect whenever $0<p<k+1$.