Thank you for bringing up this subject.

1- Experience tells us that citizens rarely care about central banks' balance sheets, if at all, even at times when the inflation rate accelerates.

2- We know that Central Banks are not there to make a profit. Thus, they should be judged based on their effectiveness in achieving stated goals. As far as sustainability is concerned, there is no real risk since most Governments guarantee the central bank's liabilities. Plus, if the source of losses is the decline in the prices of bonds that the CBs hold, then a decrease in interest rates or holding those bonds until maturity will wipe out these losses.

Sebastian Diessner, (2023), The power of folk ideas in economic policy and the central bank–commercial bank analogy, New Political Economy, 28:2, 315-328

The case of the ECB’s governance of its balance sheet reveals that although Europe’s monetary policy-makers are well aware of the irrelevance of central bank capital in theory, they nevertheless insist on maintaining positive capital in practice. When probing into the reasons behind this incongruence, the available evidence suggests that policy-makers deem themselves constrained by the general public’s misperceptions of central banking. In particular, non-experts are believed to disapprove of the central bank in case policy outcomes deviated from the ‘common sense’ that losses and negative capital are ‘not good’, in line with household reasoning.

Central banks are reporting losses due to rising interest rates which are reducing profits and even leading to losses. though losses and negative equity do not directly affect the ability of central banks to operate effectively.