The exact value conjecture for the combinatorial exceptional-point constant

Let $\delta_{\mathcal K}$ denote the universal constant for the paper's combinatorial formulation, whose associated statement is denoted by $\mathcal K_\delta$. The preceding proposition shows that if

$$(2\delta)^3+(2\delta)^2+2\delta>1,$$

then there is a counterexample to $\mathcal K_\delta$.

Exact value conjecture. The universal constant $\delta_{\mathcal K}$ is the only real root of

$$(2\delta)^3+(2\delta)^2+2\delta=1.$$

The conjecture asserts optimality of the construction giving the upper bound $\delta_{\mathcal K}<0.2719$; the paper's proof supplies the counterexamples and upper bound, while the matching optimality remains open.