All lamps on a grid

A single press toggles at most $5$ lamps. Is that enough to give a lower bound and then a matching plan?

1. A press toggles at most $5$ lamps, and $3$ presses toggle at most $15$ lamps in total, which is fewer than the $16$ lamps that must change, so at least $4$ presses are needed.
2. Number rows and columns from $1$ to $4$. Pressing the lamps $(1,3)$, $(2,1)$, $(3,4)$ and $(4,2)$, which each lie on the border and so toggle exactly $4$ lamps, toggles $16$ lamps in total.
3. These $4$ presses toggle the lamps $(1,2)$, $(1,3)$, $(1,4)$, $(2,3)$; $(1,1)$, $(2,1)$, $(2,2)$, $(3,1)$; $(2,4)$, $(3,3)$, $(3,4)$, $(4,4)$; $(3,2)$, $(4,1)$, $(4,2)$, $(4,3)$, which are all different, so every lamp is toggled once.
4. The minimum number of presses is $4$.

Answer: $4$