Would moving social security retirement to private savings cause asset price inflation?

Awhile back, I was talking about ways to fix the pay-as-you-go retirement scheme that countries have, I suggested mandatory private retirement with subsidies for the poor which would match the current budget given to social security retirement funds.

However, somebody said this would cause asset price inflation. I am somewhat skeptical of this response, specifically on how impactful it would actually be.

To my knowledge, the social security administration keeps money in US treasury bonds. My thought process is that when the social security administration privatizes retirement, the demand for treasury bonds would fall in addition to demand for other investments going up due to a private scheme in full force, meaning the asset price inflation would at best be an exaggeration since demand might shift more than it rises as a whole.

Is this thought process correct? What would be the actual impact of private savings scheme?