I mean the value of the dollar as measured by the CPI.

You do mean that, but it doesn't have a lot of bearing on the correctness of your argument. I suspect you've assumed that wealth is neither created or destroyed somewhere in your logic, because what you are saying would make sense if that was an assumption. But that would be a poor assumption if you've made it.

CPI is a tool for adjusting the value of a dollar, but it doesn't have any bearing on the value of the total economy vs the value of all the dollars in existence. All the dollars in existence might buy the entire economy many times over or they might buy a tiny fraction of it. And that ratio changes continuously (both from money creation and economic activity). There is no inverse relationship mediated by the CPI.

There are three charts on the linked website - bonds, dollars and stocks. None of them are exactly measures of wealth. My question is about how their properties as mutual numeraires could allow them to all fall at the same time.

People are willing to pay fewer dollars for each asset class.

People can't be willing to pay fewer dollars for the third asset class, which is dollars.

you absolutely can pay fewer dollars for dollars delivered at some point in the future

tautologically the price of one dollar will always be one dollar but they can be less willing to spend those dollars on anything, and simultaneously less willing to sell assets they already own for dollars.