Let $\lambda\vdash n$ denote the integer partition of $n$. Define the product $\mathcal{N}(\lambda)=\lambda_1\lambda_2\cdots\lambda_r$ when $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_r>0)$.
Let $\gamma$ be the Euler's constant. Lehmer proved that $$\lim_{n\rightarrow\infty}\frac1n\sum_{\lambda\vdash n}\frac1{\mathcal{N}(\lambda)}=e^{-\gamma}.$$
I like to ask:
QUESTION. Does the following limit exist? Or, what is the asymptotic growth of the sum? $$\lim_{n\rightarrow\infty}\sum_{\lambda\vdash n}\frac1{\mathcal{N}(\lambda)^2}.$$
ADDED. Glad to see that Fedor Petrov's answer would (potentially) work for $$\lim_{n\rightarrow\infty}\sum_{\lambda\vdash n}\frac1{\mathcal{N}(\lambda)^r}.$$