By a "homotopy commutative diagram," I mean a functor $F: \mathcal{I} \to \mathrm{Ho}(\mathrm{Top})$ to the homotopy category of spaces. By a "strictification," I mean a lifting of such a functor to the category $\mathrm{Top}$ of topological spaces. I am curious about simple instances where such a "strictification" does not exist: that is, there are obstacles to making a diagram that commutes only up to homotopy strictly commute.

For example, it is known that a homotopy coherent diagram in topological spaces can always be strictified. In other words, if one imposes homotopy commutativity, but also keeps track of all the homotopies and requires that they satisfy compatibility conditions with one another, then one can strictify. More precisely, a homotopy coherent diagram $\mathcal{I} \to \mathrm{Top}$ can be described as a simplicial functor $\mathfrak{C} \mathcal{I} \to \mathcal{Top}$, where $\mathfrak{C}\mathcal{I}$ (notation of HTT) is a "thickened" version of $\mathcal{I}$ where the usual relations in $\mathcal{I}$ only hold up to coherent homotopy. Then it is a theorem of Dwyer and Kan that the projective model structures on homotopy coherent diagrams (i.e. $\mathrm{Fun}(\mathfrak{C} \mathcal{I}, \mathrm{SSet})$) and on commutative diagrams (i.e. $\mathrm{Fun}(\mathcal{I}, \mathrm{SSet})$ are Quillen equivalent under some form of Kan extension.

Part of this question is to help myself understand why homotopy coherence is more natural than homotopy commutativity. One example I have in mind is the following: let $X$ be a connected H space, and suppose $X $ is not weakly equivalent to a loop space. Then one can construct a simplicial diagram in the homotopy category given by the nerve construction applied to $X$ (as a group object in the homotopy category). This cannot be lifted to the category of spaces, because then Segal's de-looping machinery would enable us to show that $X$ was the loop space of the geometric realization of this simplicial space.

However, this example is rather large. Is there a simple, toy example (preferably involving a finite category) of a homotopy commutative diagram that cannot be strictified?