Unassigned-outflow truncation error conjecture

Let $N_i^*$ be the truncation levels, let $\bar{t}$ be the time parameter, and let $\relax\epsilon_1(N_1^*,N_2^*,\bar{t})$ and $\relax\epsilon_2(N_1^*,N_2^*,\bar{t})$ denote the two truncation-error quantities. Error-prone truncation means $N_i^*<\underline{N}_i$, and the unassigned outflow approach is one of the truncation methods. Unassigned-outflow truncation error conjecture. In performing error-prone truncation using the unassigned outflow approach,

$$\epsilon_1(N_1^*,N_2^*,\bar{t})=0$$

and

$$\epsilon_2(N_1^*,N_2^*,\bar{t})>0.$$

Hence, it does not bias any states but cannot maintain a valid probability mass function. This conjecture concerns the trade-off between preserving the state distribution and preserving total probability mass under truncation; its resolution is not established in the supplied text.