Back-scratching scare tactics?

I’ve had 2 quotes for home insurance, one from the existing supplier, one from my bank. My current wants 13% more than last year. The bank is demanding more than the new, improved premium, and that’s after a 40% discount.
    In fact, my resident mathematician tells me that when the bank’s discount runs out next year, the bill with be double what I’m paying now.
    Which leaves me wondering if insurance providers are in cahoots. My current bungs in a demand for 13% more. A rival offers a frankly ridiculous deal – on the understanding that my current will do the same for the rival to scare existing clients to cough up and feel relieved. Financial shenannygoats in a wicked old world.