Yes, bond funds are marked to market, so they will decline as the composition of their holdings will.
Households actually have unimpressive relative levels of credit to equity holdings. The reason why is because there is little return on credit, making it irrational to hold any amount greater than to fund future liquidity needs, risk adjusted and time discounted.
The vast majority of credit is held by insurance companies. Pension funds have large stakes as well. Banks hold even fewer bonds since they try to sell them as soon as they've made them.
Insurance companies are forced to hold a large percentage of their floats in credit then preferred equity. While this dulls their returns, it's not a large problem for them because they typically hold bonds until maturity. Only the ones who misprice the risk of insurance will have to sell at unfavorable prices.
Being able to predict interest rates thus bond prices accurately would make one the best bond manager in the world. While it does look like inflation will rise again soon just as it has during every other US expansion, can it be assured when commodity prices are high in real terms and look like they may be in a collapse? The banking industry would have to produce credit at a much higher rate to counter the deflation of all physical goods.
Households typically shun assets at low prices to pursue others at high prices, so their holdings of bonds ETFs should be expected to decline during a bond collapse. If insurance companies find it less costly to hold ETFs then they will contribute to an increase in bond ETF supply.