The key idea is that we can label every uncolored segment with a positive number so that "Ptolemy is satisfied everywhere." In other words, if $ABCD$ is a convex quadrilateral formed by some four of the $n-$gon's vertices, then $\ell(AB) \cdot \ell(CD) + \ell(AD) \cdot \ell(BC) = \ell(AC) \cdot \ell(BD),$ where $\ell(s)$ denotes the label on segment $s$. Call the coloring Ptolemaic if this is the case. We claim that there is a Ptolemaic coloring of the $n-$gon's sides and diagonals with positive numbers which satisfies the $2n-3$ "given conditions." This would clearly solve the problem, since every segment $s$ is either colored with $\ell(s)$ or is not colored at all at any given point in time, where $\ell(s)$ is a constant independent of time. So let's prove this. Let $VA_1A_2 \cdots A_{n-1}$ be our polygon, where $V$ is the vertex shared by the diagonals. Let $x_1, x_2, \cdots, x_{n-1}, y_1, y_2, \cdots, y_{n-2}$ denote $\ell(VA_1), \ell(VA_2), \cdots, \ell(VA_{n-1}), \ell(A_1A_2), \ell(A_2A_3), \cdots, \ell(A_{n-2}A_{n-1})$, respectively. Let $z_1, z_2, \cdots, z_{n-1}$ denote the quantities $\frac{y_1}{x_1x_2}, \frac{y_2}{x_2x_3}, \cdots, \frac{y_{n-2}}{x_{n-2}x_{n-1}}$ respectively. The critical claim is the following. Claim. If we let $\ell(A_iA_j) = x_i x_j \cdot (z_i + z_{i+1} + \cdots + z_{j-1})$ for all $1 \le i < j \le n-1$, then the coloring is Ptolemaic. Proof. This is an easy computation, just check that all quadrilaterals satisfy the Ptolemy condition in two cases on whether or not $V$ is a vertex of the quadrilateral. Left as an exercise to the reader . $\blacksquare$ We are done! $\square$ Remark. The motivation for this coloring comes from experimentation with small $n$ .