Zachary Hess sets up a set of cases which are apparently meant to says something about whether 'kepeken' is a transitive verb (takes 'e' with its object) or a preposition (doesn't). It is
'mi XY e Z' Immediately we can see that 'XY e Z' is a predicate and that XY is a transitive verb phrase and Z is a noun phrases, the direct object. From that it follows that X is a transitive verb and Y is a modifier (adverb) – unless X is a preposition, in which case, Y might be a noun phrase, an object. Hess offers two sets of cases, each containing sentences replacing X by a prepositional 'kepeken' and by 'kepeken e' and by a blank. One set replaces Y by 'lili' and Z by 'ilo'; the other Y by 'jaki ala' and Z by 'tomo telo', both fitting into the expected categories. Prepositional 'kepeken' also can be a causative verb, with or without its own proper object.
The simplest cases, where X is empty, immediately break this”pattern” (show it is not a genuine matrix) since tp does not have a null verb (pace Lope, who sometimes seems to say that there is an invisible copula, a “be”). But he proceeds
mi lili e ilo.
I shrink the tool.
No problem here, except for not fitting the pattern. Of course, I t could also mean “I reduce the number of tools” but that is not the point here. That is, a modifier can be used as a transitive (causative) verb.
mi jaki ala e tomo telo.
I clean the bathroom.
Well, the natural reading ('ala' after the verb is sentential negation) is “I don't dirty the bathroom” However, the condition of the matrix is that 'jaki ala' is a unit, so either the complement of 'jaki', “not dirty”, which just gets things away from dirty, or the polar opposite, “clean”, Hess's intention. Again, a modifier – this time complex – can be a causative transitive verb. The various merely semantic issues can be ironed out by context (or don't really matter).
So, suppose we put prepositional 'kepeken' in for X:
mi kepeken lili e ilo.
I use that tool a little.
mi kepeken jaki ala e tomo telo.
I use the shower to clean.
Note in the first that the translation does not account for the transitive/causativity of 'kepeken' here. This translation apply directly to 'mi kepeken lili ilo'. What this says is “I cause the tool to use (something) a little' (keeping 'lili' as a modifier) or (with 'lili' as a noun, object) “I cause the tool to use smallness/reduction” Alternatively, we could chalk this up to a slip, using verbal 'kepeken' in the slot
The translation of the second is harder to account for, except that it seems to use verbal 'kepeken' again. Literally, it says “I cause the bathroom to use cleanliness/action of cleaning” or “I cause the bathroom to use (something) non-dirtily/cleanly” It's not clear where the “to clean (myself?)” ('tawa pona sama'?) come from.
Finally, the cases where 'kepeken e' are put in for X. This is illegitimate, of course, because X has to be a transitve verb and 'kepeken e' (for all we use it as a shorthand to describe the transitive forms of words) is not one, or, indeed, a verb at all, but a verb followed by the DO marker. But let's see what Hess suggests come out of this:
mi kepeken e lili e ilo.
I use small and I use a tool.
mi kepeken e jaki ala e tomo telo.
I use clean and I use bathroom.
No problems here (the modifiers as nouns might mean the property or something with the property or the action of the corresponding verb, but probably not just the modifier itself, but that is style and context). But they don't fit the matrix. Nor does all of this discussion seem to have any bearing on whether 'kepeken' needs 'e' or not.