The way you begin your question hints at a problem we'll get to below, but before that, let me suggest a distinction. It's one thing to presuppose or assume something; it's another thing for it to be a matter of convention.
There's a lot to be said on the matter of convention; there's not just one idea under that umbrella. But let's take an example from philosophy of space and time. Adolph Grünbaum argued many years ago that given our usual view of space and time, lengths are a matter of convention. The gist of the idea was this: if space and time are continuous, then any two lines contain the same number of points. In Grünbaum's view, that meant there is nothing in space and time themselves to ground the difference between different possible standards for assigning lengths. We have to pick one (think of it as deciding what counts as a ruler) and only after we've done that do questions about lengths have answers. If Grünbaum were right (I'm not convinced, but that's not our issue), then there would be no hard fact of the matter about whether two non-overlapping lines have the same length. There would be no such fact because there'd be nothing in the world to ground it. But notice: Grünbaum's argument has nothing to do with what we presuppose or assume. It's an argument about how much structure the world really has. Grünbaum believed that it's not a matter of convention whether space and time are continuous. That's a fact about the world itself.
You might ask how we could know such a thing. That's a perfectly good question, and it might not be easy to answer. Furthermore, at some point in sorting out our beliefs, we'll have to assume that some things are true without being able to prove them (otherwise we end up in circularity or regress.) But whether I assume something without proof and whether it's true aren't the same question. (And for that matter whether I assume something without proof and whether I know it aren't the same thing. Knowing might be a matter of being connected to the facts in a reliable way. I could be connected that way whether or not I could prove it.)
This might be a good time to have a look at the way you pose your question: "Isn’t it true that ultimately all truth is conventional?" Unless I'm missing something, you're trying to show that it's true, full stop, that all truth is conventional; not that this is a convention we adopt (after all, we don't) nor something that we assume or presuppose (on the contrary: most people assume no such thing) but that this is how things really are. But if that's right, then there's at least one truth that isn't merely conventional. And in that case, the claim that all truths are conventional is false.
This might seem trivial; I don't think it is. Radical conventionalism is a thought we don't have a way to think. Once that's clear, you might start to suspect that there's a good deal more in the way of truths that we don't make.