“Biggest” coefficients of a linear combination between vectors of zeros and ones

Let $$n$$ be a positive integer. Denote by $$B_n$$ the set of $$n\times(n+1)$$-matrices of rank $$n$$ and with coefficients in $$\{0,1\}$$. I would like to measure how "complex" the coefficients of a linear combination of the columns of a matrix of $$B_n$$ can be. More precisely, I'd like to compute (or estimate the asymptotic behaviour of) $$P_n := \max \left\{ \prod_{i=1}^{n+1}|\lambda_i|,\; M\in B_n,\; \sum_{i=1}^{n+1}\lambda_i m_{\star,i} = 0,\; \lambda_1,\dots,\lambda_{n+1}\in \mathbb{Z},\; \gcd(\lambda_1,\dots,\lambda_{n+1})=1 \right\}$$

where $$m_{\star,i}$$ stands for the $$i$$-th column vector of the matrix $$M$$.

$$P_2=1$$ and $$P_3 = 2$$, and for every $$1\le k\le n-2$$ with $$\gcd(k,n-1)=1$$, $$P_n$$ is bounded below by $$(n-1)(n-(k+1))^{k}k^{n-k}$$ (indeed, setting $$v$$ the vector whose $$k$$ first coefficients are 1 and $$n-k$$ last coefficients are 0 and $$\hat e_i$$ the vector whose only 0 coefficient is at line i and whose other coefficients are 1, we have the following combination: $$(n-1) v + (n-(k+1))(\hat e_1 +\dots + \hat e_k) = k(\hat e_{k+1} +\dots + \hat e_n).)$$