Consecutive p-smooth integers

[XYYXF]

Let's define L(n, k) as the largest prime factor of product

n*...*(n+k)

of k+1 consecutive integers, starting at positive integer n.

So we have, for example,
L(4374, 1) = 7
L(48, 2) = 7
L(350, 2) = 13
L(138982582998, 2) = 103
L(61011223, 3) = 163
L(23931257472314, 3) = 631
L(1517, 4) = 41
L(3294850, 5) = 239
L(1913253200, 8) = 3499
L(8559986129664, 12) = 58393
L(48503, 14) = 379

Conjecture:
as n goes to infinity,

lim inf L(n, k) / (log n)^2 = C_k

The very rough estimates of constants C_k are:
C_1 ~ 0.0376
C_2 ~ 0.258
C_3 ~ 0.907
C_4 ~ 2.46
C_5 ~ 5.16
C_6 ~ 11.4
C_7 ~ 19
C_8 ~ 42
C_9 ~ 70
C_10 ~ 140
C_11 ~ 200
C_12 ~ 250
C_13 ~ 380
C_14 ~ 430
C_15 ~ 460

Some successive minima of L(n, k) are shown there:
http://oeis.org/A193943
http://oeis.org/A193944
http://oeis.org/A193945
http://oeis.org/A193946
http://oeis.org/A193947
http://oeis.org/A193948
http://oeis.org/A199407
http://oeis.org/A200566
http://oeis.org/A200567
http://oeis.org/A200568
http://oeis.org/A200569
http://oeis.org/A200570
http://oeis.org/A209837
http://oeis.org/A209838
http://oeis.org/A209839

Any suggestions on the conjecture? Does it depend on other
known ones like Twin prime conjecture or ABC conjecture?

Great thanks for any information on the subject.