Conjecture on partitioning a dense graph into two dense spanning subgraphs

Let $s,t\geq 0$ be real numbers, and let $G$ be a non-null graph. A subgraph is $s$-dense if it has at least $s$ edges per vertex.

Partition conjecture. If

$$e(G)\geq (s+t+1)v(G),$$

then there exist vertex-disjoint non-null subgraphs $G_1$ and $G_2$ such that

$$e(G_1)\geq sv(G_1),\qquad e(G_2)\geq tv(G_2),$$

and

$$V(G_1)\cup V(G_2)=V(G).$$

The conjecture would strengthen several aspects of the paper's main theorem and implies another conjecture in the paper. The supplied text records partial results when $s=t$ or when $e(G)\geq(s+t+\frac32)v(G)$, but does not state a complete resolution.