IR-TST 2018 Problem 14

Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k$ greater than $1$ such that for all $1 \leq i, j \leq k$ , $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ Proposed by Morteza Saghafian