Personally think the currently, you can't classify current market condition as down nor bear. Look at bigger picture and long term chart, still most worldwide stock markets are not far from its historical high.
Take a look on 5 years chart, still market is at high side.
But condition can change in the future which no one can predict correctly how it will be.
Instead new fund should be launched during bear market as time and time again, it proves that bear market (especially when recession near to end) is the time to pick up some good stock that price is under distressed despite company fundamental still intact. But launching new fund during bear market, I think fund house will have difficulty to sell it, that's why nobody want to launch new fund at that time instead launching during a bull run will receive overhelming response and fully subscribe.
Yes, fund portfolios are not static throughout, but not as frequent as some retailers, buy and sell in just few days. They will review their portfolio from time to time which depends on market movement, company fundamental etc. if nothing changes much, then they generally hold for long term, only trade small portion of their fund.
Yes, if fund house launch new funds during bear market, it's really difficult to sell the funds, although that's the best time to invest. Most people will only invest when bull market and stop invest during bear market. Anyway, only people invest in other way round can be rich...