Polynomial patterns from squared fractional powers in dense sets

Let $a,b>1$ be distinct non-integers, and let $A\subseteq\mathbb{N}$ have positive upper density, meaning

$$\limsup_{N\to\infty}\frac{|A\cap\{1,\ldots,N\}|}{N}>0.$$

Multiple recurrence conjecture. The set $A$ contains a pattern of the form

$$\{m,m+\lfloor n^a\rfloor^2,m+\lfloor n^b\rfloor^2\},\qquad m,n\in\mathbb{N}.$$

This is the corresponding multiple-recurrence problem arising from the ergodic questions above and remains open.