The characterization of equality in the two-color reciprocal Rado number

Let $f_2(k)$ be the smallest positive integer $n$ such that every 2-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution, with the variables not necessarily distinct, to

$$\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_k}=\frac{1}{x_{k+1}}.$$

Call $k$ an odd prime power if $k=p^m$ for some odd prime number $p$ and positive integer $m$. Equality conjecture. If $k\geq4$ is not an odd prime power, then

$$f_2(k)=3k^2.$$

The conjecture is motivated by computational results showing that $f_2(k)=3k^2$ exactly when $k$ is an odd prime power for $k\in\{3,4,\ldots,25\}$, together with the proved result that $f_2(k)=3k^2$ when $k=3\cdot2^m$ for a positive integer $m$ and that $f_2(k)\geq3k^2+1$ for odd prime powers.