The two-periodic barcode-density conjecture

For $t\in\mathbb{Z}$ and $M\in\mathbb{Z}$, define the two truncation limits $\rho_0(t)$ and $\rho_1(t)$ by the displayed sums over $n\in\frac12\mathbb{Z}_{\ge0}$. Let $\rho^{\mathrm{barcode}}_{(L)}(t)$ denote the pre-limit barcode density. Barcode-density conjecture. The limits $\rho_0(t)$ and $\rho_1(t)$ exist for all $t\in\mathbb{Z}$, take the two parity-dependent values $\rho^{\mathrm{barcode}}_{\mathrm{even}}$ and $\rho^{\mathrm{barcode}}_{\mathrm{odd}}$ specified by the relations in the statement, and satisfy $\rho^{\mathrm{barcode}}_{\mathrm{even}}+\rho^{\mathrm{barcode}}_{\mathrm{odd}}=1$. Moreover, $\rho^{\mathrm{barcode}}_{(L)}(t)$ converges to the corresponding even or odd density according to the parity of $t$. This predicts period-two translation behavior and global density $1/2$ for the limiting barcode process; it is motivated by numerical checks, while the relevant divergent-series regularizations remain unproved.