So you basically don't know
Some private money managers (read not registered as money managers at all) could in fact be trying their luck with Bitcoin (personally, I have no doubts about that), but how that would be different from just wealthy people doing the same on their own? The question is about funds which are managing money of many thousands of investors, and this is a completely different story, since such funds are strictly monitored by supervising and controlling government agencies. Apart from that, they all have an investment policy which determines in which assets the fund managers are allowed to invest the money of their clients, and thus investing in Bitcoin might not (or rather should not) be an option for these funds (since Bitcoin is likely not allowed to be invested in)
This is why a bitcoin ETF is so important. If the SEC approves of a BTC ETF, then there will be a dramatic increase in demand for bitcoin as more and more investment funds are allowed to buy bitcoins.
I certainly agree that the price will shoot up quite nicely
But I don't think that it will shoot up specifically because there will be a great demand for bitcoins due to money being poured into Bitcoin by investment funds or just wealthy individual investors. The price surge will be primarily thanks to the hype and hysteria that will arise and reach unprecedented levels due to this approval as such (if it is ever going to happen in the first place). Many wealthy investors can invest in bitcoins directly, and if they haven't done so already (apart from a few random dudes like the Winklevii twins), there is no reason to think that investment funds, which have stricter investment policies overall, are going to invest in Bitcoin at all