Almost regular subhypergraph of hypergraph with large minimal degree
Unfortunately the conjecture is wrong (for $d \geq 2$). Here is a counterexample for $d=2$.

Suppose that the conjecture (in its graphical formulation) held for some $c,M_0 &gt; 0$. Consider a complete bipartite graph in which the left side has $M^2$ vertices and the right side has $M$ vertices. Thus the left degrees are $M$ and the right degrees are $M^2$. In order to create a subgraph in which all right degrees are at most $M$, we need to remove at least $M(M^2-M)$ edges. On the other hand, any subgraph in which all left degrees are at least $cM$ must have at most $(1-c)M^3$ edges removed. This means that $M^3 (1-1/M) \leq (1-c)M^3$, which is false for large enough $M$.