Cubic formula conjecture for the minimum number of two-by-two spectral radii

For a finite set $A\subset\mathbb{R}^+$, define

$$\Omega_2(A)=\{\rho(M):M\in A^{2\times 2}\}.$$

For $k\geq 1$, define

$$r_k:=\min\{|\Omega_2(A)|:|A|=k\}.$$

A cubic minimum conjecture asserts

$$r_k=\frac{1}{2}(2k^3-k^2+k)=kT(k)+(k-1)T(k-1),$$

where $T(k)=k(k+1)/2$ is the $k$-th triangular number; moreover, $r_k=|\Omega_2(A)|$ when $A$ is a geometric progression with $|A|=k$.