No, they won't annep1. My rent is over the Local Housing Allowance for my area. The LHA is £581.07. If I had no income at all and claimed UC, the government would pay that amount and no more. That means I would have to find the difference (£243.93) out of my own resources. If I were to apply for UC, I wouldn't receive anything for "living" (food, utility bills) because I already receive more than £73pw from occupational pensions. 0 minus £243.93 = -£243.93. Therefore, I am forced to work, which means that I'm not eligible for UC. Hope that makes sense. I've played around with benefit calculators for hours and there is no solution.
I wish people actually understood how the benefit system actually works, when they bleat on about "work always paying" and people on benefits being feckless and lazy. Boris Johnson obviously doesn't and I'm afraid to say I don't think most Labour MPs do either. For example, raising the minimum wage means a reduction in UC and means the self-employed are even less likely to be eligible.
I have never heard any politician mention Local Housing Allowance, which, for those who have first hand experience of benefits, is one of the main reasons for poverty.