gr.group theory - Collapsible group words

What is the length $f(n)$ of the shortest nontrivial group word $w_n$ in $x_1,ldots,x_n$ that collapses to $1$ when we substitute $x_i=1$ for any $i$?

For example, $f(2)=4$, with the commutator $[x_1,x_2]=x_1 x_2 x_1^{-1} x_2^{-1}$ attaining the bound.

For any $m,n ge 1$, the construction $w_{m+n}(vec{x},vec{y}):=[w_m(vec{x}),w_n(vec{y})]$ shows that $f(m+n) le 2 f(m) + 2 f(n)$.

Is $f(1),f(2),ldots$ the same as sequence A073121:
$$1,4,10,16,28,40,52,64,88,112,136,ldots ?$$

Motivation: Beating the iterated commutator construction would improve the best known bounds in size of the smallest group not satisfying an identity.

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