I agree that it's possible; indeed, I think it's easier to interpret some students as the object in this sentence than it is in my sentence. I can't yet come up with a theory why this might be. Can you?
This problem is above my syntactic pay grade, to tell you the truth. I really can't say. That said, I shall hazard a guess. Maybe we're dealing with extraposed infinitival relative clauses. Consider this example, which I am straining to produce:
Many utensils are in the drawer to eat with.
I have mixed feelings about that sentence. Do you find it grammatical? I'm not sure whether I do or not. But if it is grammatical, it seems to me that "to eat with" may indeed have the status of an infinitival relative clause:
Many utensils are in the drawer with which to eat.
The above transformation teases out the concealed relative pronoun in the first example, if indeed there is a concealed relative pronoun there. My idea is that this would derive from:
Many utensils to eat with are in the drawer.
Many utensils with which to eat are in the drawer.
The infinitival relative would then be extraposed from the noun it modifies ("utensils"), appearing instead the end of the sentence, in the same way that finite relative clauses can sometimes be extraposed.
Many utensils which you can eat with are in the drawer.
Many utensils are in the drawer which you can eat with.
Many utensils with which you can eat are in the drawer.
Many utensils are in the drawer with which you can eat.
Now, if all this is sound so far (a big "if"), it seems to me that perhaps the ease with which the related "there"-sentence could be processed has something to do with the
closer proximity of the extraposed relative to the noun modified.
There are many utensils in the drawer to eat with.
There are many utensils in the drawer with which to eat.
There are many utensils in the drawer which you can eat with.
There are many utensils in the drawer with which you can eat.