No, even with HFT the instantaneous liquidity of the market is firmly in the volatile regime, by orders of magnitude. So while HFT adds some liquidity, its very far off from bringing anything like enough to make markets more stable and it does it at the cost of many more destabilizing trades into the system.

If you look at countries such as Kenya which have huge amounts of brokerage fees (2%), which you could model a tobin tax like this on.. they have 2-9% daily swings the average volatility is much, much higher. This goes for similar exchanges with larger fees.. (or larger minimums to exchange financial instruments)

Don't believe me? Look over historical data or even just yesterdays or todays (which I wont be able to see yet when im asleep) gains or loses : https://www.nse.co.ke

Also in case you think its biased against the top or bottom movers, keep in mind there's barely 60 companies listed there.

It might be worth pointing out too, that volatility has increased on italian instruments that bear this tax, not reduced it. It's also made it harder to judge the depth of the market in terms of how much can be sold/bought/raised/invested.