Novel-partition approximation conjecture for Bernoulli singularity

Let $M=M_n$ be a random $n\times n$ Bernoulli matrix, let $S$ be the event that $M$ is singular, and let $\lambda(1),\lambda(2),\ldots$ be any enumeration of the set of novel integer partitions. For each partition $\lambda$, let $D_\lambda$ denote the event that $M$ has a right or left null vector of template $\lambda$. Novel-partition approximation conjecture. For every $r>0$, there exists $K>0$ such that

$$\mathbb{P}\left(S\setminus\bigcup_{i=1}^{K}D_{\lambda(i)}\right)=o(r^n).$$

Novel partitions are intended to provide a minimal family of templates capable of detecting singularity. The conjecture strengthens the finite-stage expansion by asserting that finitely many novel templates capture singularity up to an arbitrarily small exponential scale. The source establishes sufficiency and minimality properties of novel partitions, but not this quantitative approximation statement.