Completeness conjecture for zeroth-order conservation-law extensions of two-dimensional shallow water equations

Consider the class of two-dimensional shallow water equations with variable bottom topography, denoted by $\mathcal L$, and its spaces of zeroth-order conservation laws. Two cases are equivalent when they are related by an admissible transformation of the class. Completeness conjecture. A complete list of inequivalent cases in which the space of zeroth-order conservation laws extends is exhausted by Cases $1$, $2$, $3a$, $4_{\delta=0}$, $5$, $6$, $7a$, $8a_{\delta=0}$, $9$, $10_{\delta=0}$, $11$, $12_{\delta=0}$, $13$ and $14a$ of the classification theorem. The conjecture transfers the previously formulated group-classification conjecture to zeroth-order conservation laws; the cited classification results establish the listed cases, while completeness up to all admissible transformations remains the asserted point.