Fixed Income immunization strategies; convexity higher or lower?

In answering mocks and questions online, I run into this a lot and get confused often.

When given the choice of several portfolios to match/immunize a liability, and given details on their PV, Duration and convexity, what is the better choice for convexity? Should it be higher or lower?

On a graph with Bond price on the y-axis and interest rates on the x-axis, higher convexity means a more favorable outcome when interest rates move to the left or to the right - the 2018 past paper had an answer which favored a lower convexity.

So higher convexity is often prefereed when you immunize multiple liabilities.

For single, only ensure their Mac Durations are equal and ensure value of assets are = or close to that of the liabilities.

Since, multiple liabilities will be coming due on different times (that is they are dispersed over a time horizon) we design our asset portfolio which is dispersed as well. Dispersion is directly proportional to convexity … higher the dispersion of CFs, higher the convexity. Hence a more dispersed portfolio is better to hedge multiple liabilities.

Reverse is true for single liability

I hope i am able to explain properly