SQL Window Functions - Pivot on a Column

I have a table data as show below.

and I am expecting the below result. My data has around 1M customer_ids. Trying to implement SCD type 2 style.

I have tried below query but I am missing something that I am not getting the required answer.

select 
  cust_id,
  max(case when city_type='Physical' then city_name end) as physical_city,
  max(case when city_type='Office' then city_name end) as office_city,
  start_date,
  coalesce(lead(start_date) over(partition by party_id order by start_date), '9999-12-31') as end_date
from table
group by cust_id, start_date

Thank you!

It is necessary to consider cases when the data one type ('office' or 'physical') change 2 or more times in a row sequence ordered by start_date. For example:

After grouping (pivot) we have

So, calculation's like lead(start_date...) not enough.

1. Take ranges for every city_type start_date - to_date. We use there partition by cust_id,city_type

select *
  ,lead(start_date,1,cast('9999-12-31' as date))
    over(partition by cust_id,city_type order by start_date) to_date
From test

2. Group data by start_date (pivot). We take city_name or null for this date.
3. JOIN ranged_data to cover null's

Test data (expanded)
cust_id=1 - your example
cust_id=2 - 'office' city change twice
cust_id=3 - 'physical' city change twice
cust_id=4,5 - first rows for 'physical' and 'office' do nit have same date

See example

with ranged_data as(
select *
  ,lead(start_date,1,cast('9999-12-31' as date))
    over(partition by cust_id,city_type order by start_date) to_date
From test
)
,agg_data as(
select cust_id,start_date
  ,max(case when city_type='physical' then city_name end) physical_city
  ,max(case when city_type='office' then city_name end) office_city
from test
group by cust_id,start_date
)
select ad.cust_id,ad.start_date
  ,lead(ad.start_date,1,cast('9999-12-31' as date))
    over(partition by ad.cust_id order by ad.start_date) end_date
  ,coalesce(ad.physical_city,rd2.city_name) physical_city
  ,coalesce(ad.office_city,rd1.city_name) office_city
from agg_data ad
left join ranged_data rd1 on rd1.cust_id=ad.cust_id and rd1.city_type='office'
      and  rd1.start_date<ad.start_date and rd1.to_date>=ad.start_date
left join ranged_data rd2 on rd2.cust_id=ad.cust_id and rd2.city_type='physical'
      and  rd2.start_date<ad.start_date and rd2.to_date>=ad.start_date
order by ad.cust_id,ad.start_date

fiddle