Erdős's stronger least-common-multiple non-monotonicity conjecture

Let $C\geq 1$ be an arbitrary constant, and let $k$ be a positive integer. For integers $x$ and $y$, consider the least common multiples of the consecutive integer sets

$$\operatorname{lcm}\{x,x+1,\ldots,x+k-1\}$$

and

$$\operatorname{lcm}\{y,y+1,\ldots,y+k+C-1\}.$$

Erdős's stronger conjecture. There exist integers $k$ and $0<x<y$ with $y>x+k$ such that

$$\operatorname{lcm}\{x,x+1,\ldots,x+k-1\}>\operatorname{lcm}\{y,y+1,\ldots,y+k+C-1\}.$$

The conjecture strengthens the proved result by requiring the comparison to hold for an arbitrary fixed $C$ while asserting the existence of a suitable $k$, rather than allowing an arbitrary multiplicative factor on the right-hand side. The supplied source does not establish this stronger form; the stated reduction is proved, but the conjecture itself remains unresolved in the provided text.