Instability conjecture for the number of models of a theory

Let $T$ be a theory, let $\theta$ and $\theta$-related cardinals satisfy the displayed cardinal-arithmetic hypotheses below, and let $dot I(\lambda,T)$ denote the number of models of $T$ of cardinality $\lambda$ up to the equivalence considered in the source. A theory is definably stable when it has the corresponding stability property; here $\operatorname{cf}(\lambda)$ denotes the cofinality of $\lambda$.

Instability conjecture. If $\lambda>|T|$ is strong limit, $\lambda>\theta>\operatorname{cf}(\lambda)$, and $T$ is not definably stable, then

$$\dot I(\lambda,T)>1.$$

The same conclusion should hold under the weaker cardinal assumption $2^\theta<\theta<\lambda^\theta$, and, moreover, in that case one should have

$$\dot I(\lambda,T)\geq\left|\{\partial:\theta\leq\partial,\ 2^\partial\leq\lambda\text{ and }\partial=\operatorname{cf}(\gamma)\}\right|+1.$$

These assertions concern the model spectrum of non-definably-stable theories and strengthen the claim that such theories have more than one model in suitable cardinalities. The supplied span is marked as unresolved, and the surrounding text gives no further resolution or precise definition of every notation used in the span.