nt.number theory - Lower bounds for linear forms of logarithms (a la Baker)?

Let $lambda_1$, $lambda_2$, and $a$ be three fixed complex algebraic
numbers.

For a given integer $n$, write
$Theta(n) = arg(a lambda_1^n + lambda_2^n)$.

Assuming $Theta(n)$ is not zero, I am looking for 'good' lower bounds on
$|Theta(n)|$. By 'good' I mean that if $|Theta(n)| > B(n)$,
then $1/B(n)$ should asymptotically grow slower that any exponential in $n$.

Is there a way to use one of Baker's theorems (which provide effective
lower bounds on linear combinations of logs of algebraic numbers) to achieve
this?

For example, writing instead $Gamma(n) = arg(a lambda_1^n lambda_2^n)$
(say), one can get polynomial bounds on $|Gamma(n)|$: noting that
$displaystyle{Gamma(n) = logleft(frac{a lambda_1^n lambda_2^n}
{|a lambda_1^n lambda_2^n|}right) =
log a + n log lambda_1 + n log lambda_2 - log |a| -
n log |lambda_1| - n log |lambda_2|}$
we can apply e.g. Baker-Wustholz (1993) to the above linear form
and get a lower bound $|Gamma(n)| > C(n)$ (assuming that $|Gamma(n)|$ is non-zero)
such that $1/C(n)$ is in fact bounded by a fixed polynomial in $n$.

The problem in getting a similar lower bound for $|Theta(n)|$ is that, even though
$Theta(n)$ can be written as a linear combination of logs of algebraic
numbers of constant degree, as for $Gamma(n)$, the height of the
algebraic number $a lambda_1^n + lambda_2^n$ is potentially exponential in
$n$, and it does not seem that taking logs will help here.

The critical case is of course when $lambda_1$ and $lambda_2$ have the
same magnitude. In fact, I would be happy for an approach with even very
simple values of $a$, such as $a = 2$.