The exponential Gaudin existence conjecture outside the Virasoro highest-weight locus

Let the exponential Gaudin Bethe Ansatz system be as in the source, and let $d_0=r^2$, $d_1=r(r-1)$ characterize the Virasoro highest-weight locus. Exponential Gaudin nonexistence conjecture. If these equalities do not hold, then the exponential Gaudin system has no solutions. The preceding proposition proves uniqueness on the stated locus; the converse nonexistence assertion is explicitly presented as an expectation and remains open in the supplied text.